Elements of Algebra

CHAP. XVIII. OF ALGEBRA. 55 ber prefixed to it, this always shews that the square-root is meant. 191. To explain this matter still better, we shall here exhibit the different roots of the number $a$, with their respective values: $ \left. \begin{array}{c} \sqrt{a} \\ \sqrt[3]{a} \\ \sqrt[4]{a} \\ \sqrt[5]{a} \\ \sqrt[6]{a} \end{array} \right\} \text{ is the } \left\{ \begin{array}{c} \text{2d} \\ \text{3d} \\ \text{4th} \\ \text{5th} \\ \text{6th} \end{array} \right\} \text{ root of } \left\{ \begin{array}{c} a, \\ a, \\ a, \\ a, \\ a, \end{array} \right. $ and so on. So that, conversely, $ \left\{ \begin{array}{c} \text{The 2d} \\ \text{The 3d} \\ \text{The 4th} \\ \text{The 5th} \\ \text{The 6th} \end{array} \right\} \text{ power of } \left\{ \begin{array}{c} \sqrt{a} \\ \sqrt[3]{a} \\ \sqrt[4]{a} \\ \sqrt[5]{a} \\ \sqrt[6]{a} \end{array} \right\} \text{ is equal to } \left\{ \begin{array}{c} a, \\ a, \\ a, \\ a, \\ a, \end{array} \right. $ and so on. 192. Whether the number $a$ therefore be great or small, we know what value to affix to all these roots of different degrees. It must be remarked also, that if we substitute unity for $a$, all those roots remain constantly $1$; because all the powers of $1$ have unity for their value. If the number $a$ be greater than $1$, all its roots will also exceed unity. Lastly, if that number be less than $1$, all its roots will also be less than unity. 193. When the number $a$ is positive, we know from what was before said of the square and cube roots, that all the other roots may also be determined, and will be real and possible numbers But if the number $a$ be negative, its second, fourth, sixth, and all its even roots, become impossible, or imaginary numbers; because all the powers of an even order, whether of positive or of negative numbers, are affected by the sign $\pm$: whereas the third, fifth, seventh, and all its odd roots, become negative, but rational; because the odd powers of negative numbers are also negative. 194. We have here also an inexhaustible source of new kinds of surds, or irrational quantities; for whenever the number $a$ is not really such a power, as some one of the foregoing indices represents, or seems to require, it is impossible to express that root either in whole numbers or in fractions; and, consequently, it must be classed among the numbers which are called irrational.